integral

In general, the integral of a real-valued function \[f(x)\] with respect to a real value \[x\] on an interval \[[a,b]\] is written as \[\int_{a}^{b}f(x)\,dx\].

• The symbol \[dx\] represents the differential (infinitesimal changes) of \[x\]
  
• The function \[f(x)\] is known as the integrand
  
• \[[a,b]\] represents the interval of integration
  

When limits are omitted, \[\int f(x)\,dx\], the integral is called an indefinite integral, which represents the antiderivative whose deriative is the integrand, \[f(x)\].

Integrals (or definite integrals) should not be confused with antiderivatives (indefinite integrals), as integration is used to find the area (essentially just the difference between \[F(b)-F(a)\]), while antidifferentiation is used to calculate \[\int f(x)\,dx=F(x)\]. Thus, the calculation of \[\int_{a}^{b}f(x)\,dx\] is a two-step process, involving antidifferentiation followed by integration.

Riemann integral is one of the most commonly used definition for integrals. The definition for \[\int_{a}^{b}f(x)\,dx\] is the limit \[\lim_{\Vert \Delta x_{i} \Vert\to 0}\sum_{i=1}^{n}f \left( x_{i}^{\ast} \right)\Delta x_{i}\], where \[\Vert \Delta x_{i} \Vert\] refers to the mesh (the length of the largest sub-interval).

The Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval. A tagged partition of a closed interval \[[a,b]\] on the real line is a finite sequence, \[a=x_{0}\le x_{1}^{\ast}\le x_{1}\le x_{2}^{\ast}\le x_{2}\le\cdots\le x_{n-1}\le x_{n}^{\ast}\le x_{n}=b\], where \[x_{i}^{\ast}\] represents the type of Riemann sum (e.g. \[x_{i}^{\ast}=x_{i-1}\] for left Riemann sum) and \[\Delta x_{i}=x_{i}-x_{i-1}\].

• Riemann integral